Default time ($\tau$)

The random time at which credit default occurs. In an observed default model it is a stopping time for the market filtration.

In a deterministic-intensity model, $\lambda (t)$ is the instantaneous conditional default rate given survival and $G(t)$ is the survival probability. Risk-neutral intensity is calibrated for pricing and can differ from the physical intensity used for forecasting losses. More general stochastic-intensity models condition on the underlying filtration and require joint expectations with discounting and recovery.